Integrate a nonuniform permittivity through the gap
A spatially varying dielectric does not generally act like its arithmetic-average permittivity. In this one-dimensional ideal gap, ε(z)=ε0(1+αz/d), there is no free volume charge and the normal displacement D is constant. The electric field is E(z)=D/ε(z), so it decreases where the permittivity is larger.
Learning outcome: Compare α=0,1,2,4 at fixed voltage. Retain constant D, varying boundary fields and both effective and arithmetic-mean factors. Verify the α=0 limit separately and explain why rearranging the same materials side by side changes the averaging law.
Starting recap
Review charge conservation, conductor equipotential conditions, definite field integrals and energy derivatives under a stated electrical constraint. Convert capacitance and charge units before interpreting small energies.
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For α=2, the effective relative factor is 2/ln3≈1.820478, not the arithmetic mean 2. With area 0.01 m², gap 0.001 m and V=6 V, the first-boundary field is 6000(2/ln3) N/C and the second-boundary field is one third of it. D is the same at both boundaries.
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For α=2, which effective relative factor follows from the voltage integral?
Experiment
Compare α=0,1,2,4 at fixed voltage. Retain constant D, varying boundary fields and both effective and arithmetic-mean factors. Verify the α=0 limit separately and explain why rearranging the same materials side by side changes the averaging law.
Integrate a spatially graded permittivity
Worked teaching model · synthetic values · Ideal planar area .01 m², gap .001 m, linear dielectric epsilon(z)=epsilon0(1+alpha z/d), with no free volume charge in the gap. One-dimensional no-fringe approximation. Displacement D is constant, E is not. Linear lossless local constitutive law is stipulated, not a real material recommendation.
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Nonuniform electric field with constant displacement
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: D divided by local epsilon (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Nonuniform electric field with constant displacement
D divided by local epsilon
| fractional gap coordinate | electric field (N/C) |
|---|---|
| 0 | 10922.87 |
| 0.0125 | 10656.46 |
| 0.025 | 10402.73 |
| 0.0375 | 10160.81 |
| 0.05 | 9929.882 |
| 0.0625 | 9709.218 |
| 0.075 | 9498.148 |
| 0.0875 | 9296.06 |
| 0.1 | 9102.392 |
| 0.1125 | 8916.629 |
| 0.125 | 8738.297 |
| 0.1375 | 8566.957 |
| 0.15 | 8402.208 |
| 0.1625 | 8243.676 |
| 0.175 | 8091.015 |
| 0.1875 | 7943.906 |
| 0.2 | 7802.051 |
| 0.2125 | 7665.172 |
| 0.225 | 7533.014 |
| 0.2375 | 7405.336 |
| 0.25 | 7281.914 |
| 0.2625 | 7162.538 |
| 0.275 | 7047.013 |
| 0.2875 | 6935.156 |
| 0.3 | 6826.794 |
| 0.3125 | 6721.767 |
| 0.325 | 6619.922 |
| 0.3375 | 6521.117 |
| 0.35 | 6425.218 |
| 0.3625 | 6332.099 |
| 0.375 | 6241.64 |
| 0.3875 | 6153.73 |
| 0.4 | 6068.262 |
| 0.4125 | 5985.135 |
| 0.425 | 5904.254 |
| 0.4375 | 5825.531 |
| 0.45 | 5748.879 |
| 0.4625 | 5674.219 |
| 0.475 | 5601.472 |
| 0.4875 | 5530.567 |
| 0.5 | 5461.435 |
| 0.5125 | 5394.01 |
| 0.525 | 5328.23 |
| 0.5375 | 5264.034 |
| 0.55 | 5201.367 |
| 0.5625 | 5140.174 |
| 0.575 | 5080.405 |
| 0.5875 | 5022.01 |
| 0.6 | 4964.941 |
| 0.6125 | 4909.155 |
| 0.625 | 4854.609 |
| 0.6375 | 4801.262 |
| 0.65 | 4749.074 |
| 0.6625 | 4698.009 |
| 0.675 | 4648.03 |
| 0.6875 | 4599.103 |
| 0.7 | 4551.196 |
| 0.7125 | 4504.277 |
| 0.725 | 4458.315 |
| 0.7375 | 4413.281 |
| 0.75 | 4369.148 |
| 0.7625 | 4325.889 |
| 0.775 | 4283.479 |
| 0.7875 | 4241.892 |
| 0.8 | 4201.104 |
| 0.8125 | 4161.094 |
| 0.825 | 4121.838 |
| 0.8375 | 4083.316 |
| 0.85 | 4045.508 |
| 0.8625 | 4008.393 |
| 0.875 | 3971.953 |
| 0.8875 | 3936.17 |
| 0.9 | 3901.025 |
| 0.9125 | 3866.503 |
| 0.925 | 3832.586 |
| 0.9375 | 3799.259 |
| 0.95 | 3766.507 |
| 0.9625 | 3734.315 |
| 0.975 | 3702.668 |
| 0.9875 | 3671.553 |
| 1 | 3640.957 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Vacuum capacitance | 8.84194e-11 | F |
| Effective relative permittivity | 1.820478 | ratio |
| Arithmetic-mean relative permittivity | 2 | ratio |
| Graded dielectric capacitance | 1.60966e-10 | F |
| Free plate charge | 9.65794e-10 | C |
| Constant normal displacement | 9.65794e-8 | C/m² |
| Field at first boundary | 10922.87 | N/C |
| Field at second boundary | 3640.957 | N/C |
| Stored energy | 2.89738e-9 | J |
A dielectric gradient in the field direction combines through an integral of reciprocal permittivity, not its arithmetic mean.
No free volume charge gives constant normal D. Therefore V=integral D dz/epsilon(z)=Dd ln(1+alpha)/(epsilon0 alpha), with the continuous alpha=0 limit V=Dd/epsilon0. C/A=epsilon0 alpha/[d ln(1+alpha)]. The energy integral (1/2)integral E D dVolume equals CV²/2. This local linear constitutive model excludes loss, nonlinear polarization and breakdown.
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A dielectric gradient in the field direction combines through an integral of reciprocal permittivity, not its arithmetic mean.
No free volume charge gives constant normal D. Therefore V=integral D dz/epsilon(z)=Dd ln(1+alpha)/(epsilon0 alpha), with the continuous alpha=0 limit V=Dd/epsilon0. C/A=epsilon0 alpha/[d ln(1+alpha)]. The energy integral (1/2)integral E D dVolume equals CV²/2. This local linear constitutive model excludes loss, nonlinear polarization and breakdown.
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